The given function is:

$F(x, y, z) = \frac{\sqrt{xy + z + 1}}{x - 1}.$

To find the domain and range of this function, we need to examine the conditions under which this function is defined.

Domain:

1. Square Root Condition: The expression inside the square root must be non-negative: $xy + z + 1 \geq 0.$

   This implies: $z \geq -xy - 1.$

2. Denominator Condition: The denominator cannot be zero: $x - 1 \neq 0 \quad \implies \quad x \neq 1.$

Thus, the domain is: $\{(x, y, z) \in \mathbb{R}^3 \,|\, x \neq 1 \text{ and } z \geq -xy - 1\}.$

Range:

1. The numerator, $\sqrt{xy + z + 1}$, ensures that the function output is non-negative ($\geq 0$).
2. Since the denominator, $x - 1$, can take any nonzero value, the function can output both positive and negative values.
3. The range is all real numbers except possibly zero because the numerator is zero only when $xy + z + 1 = 0$, which is within the domain.

Thus, the range is: $F(x, y, z) \in \mathbb{R} \setminus \{0\}.$

Do you want me to explain any part in detail?

Here are some related questions to extend this discussion:

1. What happens to the domain if we remove the square root?
2. How does the domain change if $z$ is fixed as a constant?
3. Can the function be simplified under specific values of $x, y, z$?
4. How do you graph the region of validity of the domain in 3D space?
5. What happens to the range if $x$ is restricted to positive values only?

Tip: Always start analyzing a function by examining restrictions caused by denominators and square roots.